Constructing Auxiliary Dynamics for Nonequilibrium Stationary States by Variance Minimization

We present a strategy to construct guiding distribution functions (GDFs) based on variance minimization. Auxiliary dynamics via GDFs mitigates the exponential growth of variance as a function of bias in Monte Carlo estimators of large deviation functions. The variance minimization technique exploits the exact properties of eigenstates of the tilted operator that defines the biased dynamics in the nonequilibrium system. We demonstrate our techniques in two classes of problems. In the continuum, we show that GDFs can be optimized to study the interacting driven diffusive systems where the efficiency is systematically improved by incorporating higher correlations into the GDF. On the lattice, we use a correlator product state ansatz to study the 1D weakly asymmetric simple exclusion process. We show that with modest resources, we can capture the features of the susceptibility in large systems that mark the phase transition from uniform transport to a traveling wave state. Our work extends the repertoire of tools available to study nonequilibrium properties in realistic systems

Analysis of a parallel multigrid algorithm

The parallel multigrid algorithm of Frederickson and McBryan (1987) is considered. This algorithm uses multiple coarse-grid problems (instead of one problem) in the hope of accelerating convergence and is found to have a close relationship to traditional multigrid methods. Specifically, the parallel coarse-grid correction operator is identical to a traditional multigrid coarse-grid correction operator, except that the mixing of high and low frequencies caused by aliasing error is removed. Appropriate relaxation operators can be chosen to take advantage of this property. Comparisons between the standard multigrid and the new method are made

Importance sampling large deviations in nonequilibrium steady states. I

Large deviation functions contain information on the stability and response of systems driven into nonequilibrium steady states, and in such a way are similar to free energies for systems at equilibrium. As with equilibrium free energies, evaluating large deviation functions numerically for all but the simplest systems is difficult, because by construction they depend on exponentially rare events. In this first paper of a series, we evaluate different trajectory-based sampling methods capable of computing large deviation functions of time integrated observables within nonequilibrium steady states. We illustrate some convergence criteria and best practices using a number of different models, including a biased Brownian walker, a driven lattice gas, and a model of self-assembly. We show how two popular methods for sampling trajectory ensembles, transition path sampling and diffusion Monte Carlo, suffer from exponentially diverging correlations in trajectory space as a function of the bias parameter when estimating large deviation functions. Improving the efficiencies of these algorithms requires introducing guiding functions for the trajectories.Comment: Published in JC

Exact fluctuations of nonequilibrium steady states from approximate auxiliary dynamics

We describe a framework to significantly reduce the computational effort to evaluate large deviation functions of time integrated observables within nonequilibrium steady states. We do this by incorporating an auxiliary dynamics into trajectory based Monte Carlo calculations, through a transformation of the system's propagator using an approximate guiding function. This procedure importance samples the trajectories that most contribute to the large deviation function, mitigating the exponentially complexity of such calculations. We illustrate the method by studying driven diffusions and interacting lattice models in one and two dimensions. Our work offers an avenue to calculate large deviation functions for high dimensional systems driven far from equilibrium.Comment: Accepted in Physical Review Letters (2018). v1: Main document: 5 pages, 3 figures. Supplementary information: 5 pages, 3 figures. v2: Main document: 5 pages, 3 figures. Supplementary information: 6 pages, 3 figures. Fixed some typos and notational inconsistencies. Expanded continuum tilted operator derivation in supplementary sectio

Segmentation of the evolving left ventricle by learning the dynamics

We propose a method for recursive segmentation of the left ventricle (LV) across a temporal sequence of magnetic resonance (MR) images. The approach involves a technique for learning the LV boundary dynamics together with a particle-based inference algorithm on a loopy graphical model capturing the temporal periodicity of the heart. The dynamic system state is a low-dimensional representation of the boundary, and boundary estimation involves incorporating curve evolution into state estimation. By formulating the problem as one of state estimation, the segmentation at each particular time is based not only on the data observed at that instant, but also on predictions based on past and future boundary estimates. We assess and demonstrate the effectiveness of the proposed framework on a large data set of breath-hold cardiac MR image sequences

Hepatitis C virus infection

Hepatitis C virus (HCV) often causes persistent infection, and is an important factor in the etiology of fibrosis, cirrhosis, and hepatocellular carcinoma (HCC). There are no preventive or therapeutic vaccines available against HCV. Treatment strategies of HCV infection are likely to improve with recently discovered direct antiviral agents (DAAs). However, a proportion of patients still progress to liver failure and/or HCC despite having been cured of the infection. Thus, there is a need for early diagnosis and therapeutic modalities for HCV relatedend stage liver disease prevention. HCV genome does not integrate into its host genome, and has a predominantly cytoplasmic life cycle. Therefore, HCV mediated liver disease progression appears to involve indirect mechanisms from persistent infection of hepatocytes. Studying the underlying mechanisms of HCV mediated evasion of immune responses and liver disease progression is challenging due to the lack of a naturally susceptible small animal model. We and other investigators have used a number of experimental systems to investigate the mechanisms for establishment of chronic HCV infection and liver disease progression. HCV infection modulates immune systems. Further, HCV infection of primary human hepatocytes promotes growth, induces phenotypic changes, modulates epithelial mesenchymal transition (EMT) related genes, and generates tumor initiating stem-like cells (TISCs). HCV infection also modulates microRNAs (miRNAs), and influences growth by overriding normal death progression of primary human hepatocytes for disease pathogenesis. Understanding these observations at the molecular level should aid in developing strategies for additional effective therapies against HCV mediated liver disease progression

Constructing Auxiliary Dynamics for Nonequilibrium Stationary States by Variance Minimization

We present a strategy to construct guiding distribution functions (GDFs) based on variance minimization. Auxiliary dynamics via GDFs mitigates the exponential growth of variance as a function of bias in Monte Carlo estimators of large deviation functions. The variance minimization technique exploits the exact properties of eigenstates of the tilted operator that defines the biased dynamics in the nonequilibrium system. We demonstrate our techniques in two classes of problems. In the continuum, we show that GDFs can be optimized to study the interacting driven diffusive systems where the efficiency is systematically improved by incorporating higher correlations into the GDF. On the lattice, we use a correlator product state ansatz to study the 1D weakly asymmetric simple exclusion process. We show that with modest resources, we can capture the features of the susceptibility in large systems that mark the phase transition from uniform transport to a traveling wave state. Our work extends the repertoire of tools available to study nonequilibrium properties in realistic systems

Embedding of a pseudo-residual design into a Möbius plane

AbstractLet U be a class of subsets of a finite set X. Elements of U are called blocks. Let v, t and λ1, 0 ⩽ i ⩽ t, be nonnegative integers, and K be a subset of nonnegative integers such that every member of K is at most v. A pair (X, U) is called a (λ0, λ1,…, λt; K, υ)t-design if (1) |X| = υ, (2) every i-subset of X is contained in exactly λt blocks, 0 ⩽ i ⩽ t, and (3) for every block A in U, |A| ϵ K. It is well-known that if K consists of a singleton k, then λ0,…, λt − 1 can be determined from υ, t, k and λt. Hence, we shall denote a (λ0,…, λt; {k}, υ)t-design by Sλ(t, k, υ), where λ = λt. A Möbius plane M is an S1(3, q + 1, q2 + 1), where q is a positive integer. Let A be a fixed block in M. If A is deleted from M together with the points contained in A, then we obtain a residual design M′ with parameters λ0 = q3 + q − 1, λ1 = q2 + q, λ2 = q + 1, λ3 = 1, K = {q + 1, q, q − 1}, and υ = q2 − 1. We define a design to be a pseudo-block-residual design of order q (abbreviated by PBRD(q)) if it has these parameters. We consider the reconstruction problem of a Möbius plane from a given PBRD(q). Let B and B′ be two blocks in a residual design M′. If B and B′ are tangent to each other at a point x, and there exists a block C of size q + 1 such that C is tangent to B at x and is secant to B′, then we say B is r-tangent to B′ at x. A PBRD(q) is said to satisfy the r-tangency condition if for every block B of size q, and any two points x and y not in B, there exists at most one block which is r-tangent to B and contains x and y. We show that any PBRD(q)D can be uniquely embedded into a Möbius plane if and only if D satisfies the r-tangency condition